FOM: fom: topos theory

[Steve Awodey]

Colin McLarty has been doing such a good job of defending topos theory
against Steve Simpson's criticism that there seemed no need to jump in
(except perhaps to counter the impression that Colin is alone in his
opinions).  But the recent turn to talking past each other and ad hominem
argument prompts me to finally try to come to Colin's aid.
	In particular, with regard to Steve's posting:

>Date: Thu, 15 Jan 1998 20:41:29 -0500 (EST)
>From: Stephen G Simpson <simpson at math.psu.edu>
>Subject: FOM: topos theory qua f.o.m.; topos theory qua pure math
>
>Colin McLarty writes:
> > >I started the discussion by asking about real analysis in topos
> > >theory.  McLarty claimed that there is no problem about this.  After a
> > >lot of back and forth, it turned out that the basis of McLarty's claim
> > >is that the topos axioms plus two additional axioms give a theory that
> > >is easily intertranslatable with Zermelo set theory with bounded
> > >comprehension and choice.
> >
> >     	No, not at all. The basis of my claim was that you can do
> > real analysis in any topos with a natural number object. In that
> > generality the results are far weaker than in ZF (even without
> > the axiom of choice)--and allow many variant extensions with
> > various uses.
>
>Darn, I thought I had finally pinned you down on this.  It sounded for
>all the world as if you were saying that the axiom of choice is useful
>for real analysis in a topos.  Now I don't know what the heck you're
>saying.  I'm losing patience, but I'll try one more time.

	This suggests that Colin is trying to dodge the point - but in
fact, it seems clear to me that it's Steve who's preferring to split hairs
rather than hear what's being said; namely that there's absolutely no
difficulty involved in doing real analysis (or virtually any other modern
mathematics you choose) in a topos.  There are (interesting) differences
depending on what kind of topos one is referring to, but there is no
problem with developing analysis in the usual way in any topos with NNO, as
Colin has clearly said several times.

In the same post, Steve Simpsom continues:

> > >The two additional axioms are: (a) "there exists a natural number
> > >object (defined in terms of primitive recursion)"; (b) "every
> > >surjection has a right inverse" (i.e. the axiom of choice), which
> > >implies the law of the excluded middle.
> > >
> > >[ Question: Do (a) and (b) hold in categories of sheaves? ]
> >
> >     	(a) holds in every category of sheaves (on a topological
> > space or indeed on any Grothendieck site). (b) holds for sheaves
> > on a topological space only for a narrow range of spaces--for
> > Hausdorff spaces it holds only when the space is a a single point.
> > (Even if you look at all Grothendieck sites, you get very few
> > more toposes satisfying (b).)
>
>That's what I thought.  So sheaves may be a good motivation for topos
>theory plus (a), but they provide no motivation for topos theory plus
>(a) plus (b).
>

This is just plain wrong; any topos of sheaves on a complete boolean
algebra also satisfies (b), as does the topos of G-Sets for any group G
(these are sets equipped with an action by G; G-Sets is also a topos of
sheaves).  As Steve himself has admitted, he really doesn't know anything
about topos theory.  So why try to pronounce on it's motivation?

Again, from that posting:
> > I think this is a crucial difference between set theory and
> > categorical foundations. Set theory has ONLY a foundational role in
> > mathematics. Category theory is used in many ways.
>
>Here on the FOM list, the crucial question for topos theorists is
>whether topos theory has ANY foundational role in mathematics.  From
>the way this dialog with you is proceeding, I'm beginning to think
>that it doesn't.  Or if it does, you aren't able or willing to
>articulate it.
>

This really seems unfair.  Colin has been quite patient about going over
ground that's been well-known for literally decades.  How often do we need
to re-ask and re-answer this tired question, whether topos theory has a
foundational role in mathematics?  Maybe until the topos theorists (and the
current generation of open-minded students) get tired of the dialog and
decide to spend their energy in more productive ways?

>Topos theorists frequently make far-reaching but vague claims to the
>effect that topos theory is important for f.o.m.  I have serious
>doubts, but I don't understand topos theory well enough to know for
>sure that topos theory *isn't* important for f.o.m.  I'm giving you a
>chance to articulate why you think topos theory is important for
>f.o.m.  So far, you haven't articulated it.

I don't recall this exchange starting by Colin being given a chance to
articulate why he thinks ... it seems to me that he neither needs to be
given such a chance (given his published record), nor has he accepted any
such challenge against which his remarks here should be judged.
	In fact, I recall this particular exchange arising in a very
different way, with Steve making the following unwarranted assertion about
analysis in toposes and Colin calling him on it:

>From: Stephen G Simpson <simpson at math.psu.edu>
>Sender: owner-fom at math.psu.edu
>To: fom at math.psu.edu
>Subject: FOM: what are the most basic mathematical concepts?
>Date: Mon, 12 Jan 1998 20:00:18 -0500 (EST)
>
>Has there ever been a decent
>foundational scheme based on functions rather than sets?  I know that
>some category theorists want to claim that topos theory does this, but
>that seems pretty indefensible.  For one thing, there doesn't seem to
>be any way to do real analysis in a topos.
>
>-- Steve
>

By the way, as for the relationship between toposes in general and
categories of sheaves:

>------------------------------
>
>Date: Wed, 14 Jan 1998 12:44:19 -0500 (EST)
>From: Stephen G Simpson <simpson at math.psu.edu>
>Subject: FOM: toposes vs. categories of sheaves
>
>Colin Mclarty writes:
> > >Categories of sheaves are toposes, but the notion of topos is much
> > >more general.
> >
> > 	This is like saying Hermann Weyl's GROUP THEORY AND QUANTUM
> > MECHANICS contains no group theory, since it only concerns the
> > classical transformation groups and the abstract group concept is
> > much more general.
>
>Huh?  This strikes me as a misleading analogy.
>
>I'm no expert on toposes, so correct me if I'm wrong, but I'm pretty
>sure that the notion of topos is *much much much* more general than
>the category of sheaves on a topological space or even a poset.

You're wrong.

>For
>instance, I seem to remember that if you start with any group G acting
>on any space X, there is a topos of invariant sheaves.  So arbitrary
>groups come up right away.  And this is just one example.

I don't know what point you're trying to make about "arbitrary groups"
here, but the topos of G-sheaves you mention hardly supports the claim
"that the notion of a topos *much much much* more general than the category
of sheaves on a topological space or even a poset" - the G-sheaves _are_
sheaves on a space, and the topos of G-sheaves sits inside the topos of all
sheaves in the most transparent way.  Is it a matter of whether that counts
as *much much much* or just *much*?

>Aren't
>there lots of other toposes that have absolutely nothing to do with
>sheaves?

No.  In fact - modulo some technicalities that you surely don't care about
- every topos is a subtopos of one of the form G-sheaves for a suitable
(generalized) space and (generalized) group G.


Name: Steve Awodey
Position: Assistant Professor of Philosophy
Institution: Carnegie Mellon University
Research interest: category theory, logic, history and philosophy of math
More information: http://www.andrew.cmu.edu/user/awodey/